# Properties

 Label 2240.2.a.bd.1.1 Level $2240$ Weight $2$ Character 2240.1 Self dual yes Analytic conductor $17.886$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$2240 = 2^{6} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2240.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$17.8864900528$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 2240.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.56155 q^{3} -1.00000 q^{5} +1.00000 q^{7} +3.56155 q^{9} +O(q^{10})$$ $$q-2.56155 q^{3} -1.00000 q^{5} +1.00000 q^{7} +3.56155 q^{9} +2.56155 q^{11} -4.56155 q^{13} +2.56155 q^{15} -4.56155 q^{17} +1.12311 q^{19} -2.56155 q^{21} +5.12311 q^{23} +1.00000 q^{25} -1.43845 q^{27} +5.68466 q^{29} -6.56155 q^{33} -1.00000 q^{35} -6.00000 q^{37} +11.6847 q^{39} -3.12311 q^{41} +9.12311 q^{43} -3.56155 q^{45} -3.68466 q^{47} +1.00000 q^{49} +11.6847 q^{51} -3.12311 q^{53} -2.56155 q^{55} -2.87689 q^{57} -4.00000 q^{59} +9.36932 q^{61} +3.56155 q^{63} +4.56155 q^{65} -6.24621 q^{67} -13.1231 q^{69} -8.00000 q^{71} +4.24621 q^{73} -2.56155 q^{75} +2.56155 q^{77} +6.56155 q^{79} -7.00000 q^{81} +4.00000 q^{83} +4.56155 q^{85} -14.5616 q^{87} +7.12311 q^{89} -4.56155 q^{91} -1.12311 q^{95} -14.8078 q^{97} +9.12311 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} - 2q^{5} + 2q^{7} + 3q^{9} + O(q^{10})$$ $$2q - q^{3} - 2q^{5} + 2q^{7} + 3q^{9} + q^{11} - 5q^{13} + q^{15} - 5q^{17} - 6q^{19} - q^{21} + 2q^{23} + 2q^{25} - 7q^{27} - q^{29} - 9q^{33} - 2q^{35} - 12q^{37} + 11q^{39} + 2q^{41} + 10q^{43} - 3q^{45} + 5q^{47} + 2q^{49} + 11q^{51} + 2q^{53} - q^{55} - 14q^{57} - 8q^{59} - 6q^{61} + 3q^{63} + 5q^{65} + 4q^{67} - 18q^{69} - 16q^{71} - 8q^{73} - q^{75} + q^{77} + 9q^{79} - 14q^{81} + 8q^{83} + 5q^{85} - 25q^{87} + 6q^{89} - 5q^{91} + 6q^{95} - 9q^{97} + 10q^{99} + O(q^{100})$$

## 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
$$3$$ −2.56155 −1.47891 −0.739457 0.673204i $$-0.764917\pi$$
−0.739457 + 0.673204i $$0.764917\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 3.56155 1.18718
$$10$$ 0 0
$$11$$ 2.56155 0.772337 0.386169 0.922428i $$-0.373798\pi$$
0.386169 + 0.922428i $$0.373798\pi$$
$$12$$ 0 0
$$13$$ −4.56155 −1.26515 −0.632574 0.774500i $$-0.718001\pi$$
−0.632574 + 0.774500i $$0.718001\pi$$
$$14$$ 0 0
$$15$$ 2.56155 0.661390
$$16$$ 0 0
$$17$$ −4.56155 −1.10634 −0.553170 0.833069i $$-0.686582\pi$$
−0.553170 + 0.833069i $$0.686582\pi$$
$$18$$ 0 0
$$19$$ 1.12311 0.257658 0.128829 0.991667i $$-0.458878\pi$$
0.128829 + 0.991667i $$0.458878\pi$$
$$20$$ 0 0
$$21$$ −2.56155 −0.558977
$$22$$ 0 0
$$23$$ 5.12311 1.06824 0.534121 0.845408i $$-0.320643\pi$$
0.534121 + 0.845408i $$0.320643\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.43845 −0.276829
$$28$$ 0 0
$$29$$ 5.68466 1.05561 0.527807 0.849364i $$-0.323014\pi$$
0.527807 + 0.849364i $$0.323014\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ −6.56155 −1.14222
$$34$$ 0 0
$$35$$ −1.00000 −0.169031
$$36$$ 0 0
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 0 0
$$39$$ 11.6847 1.87104
$$40$$ 0 0
$$41$$ −3.12311 −0.487747 −0.243874 0.969807i $$-0.578418\pi$$
−0.243874 + 0.969807i $$0.578418\pi$$
$$42$$ 0 0
$$43$$ 9.12311 1.39126 0.695630 0.718400i $$-0.255125\pi$$
0.695630 + 0.718400i $$0.255125\pi$$
$$44$$ 0 0
$$45$$ −3.56155 −0.530925
$$46$$ 0 0
$$47$$ −3.68466 −0.537463 −0.268731 0.963215i $$-0.586604\pi$$
−0.268731 + 0.963215i $$0.586604\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 11.6847 1.63618
$$52$$ 0 0
$$53$$ −3.12311 −0.428992 −0.214496 0.976725i $$-0.568811\pi$$
−0.214496 + 0.976725i $$0.568811\pi$$
$$54$$ 0 0
$$55$$ −2.56155 −0.345400
$$56$$ 0 0
$$57$$ −2.87689 −0.381054
$$58$$ 0 0
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 9.36932 1.19962 0.599809 0.800143i $$-0.295243\pi$$
0.599809 + 0.800143i $$0.295243\pi$$
$$62$$ 0 0
$$63$$ 3.56155 0.448713
$$64$$ 0 0
$$65$$ 4.56155 0.565791
$$66$$ 0 0
$$67$$ −6.24621 −0.763096 −0.381548 0.924349i $$-0.624609\pi$$
−0.381548 + 0.924349i $$0.624609\pi$$
$$68$$ 0 0
$$69$$ −13.1231 −1.57984
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ 4.24621 0.496981 0.248491 0.968634i $$-0.420065\pi$$
0.248491 + 0.968634i $$0.420065\pi$$
$$74$$ 0 0
$$75$$ −2.56155 −0.295783
$$76$$ 0 0
$$77$$ 2.56155 0.291916
$$78$$ 0 0
$$79$$ 6.56155 0.738232 0.369116 0.929383i $$-0.379660\pi$$
0.369116 + 0.929383i $$0.379660\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ 4.56155 0.494770
$$86$$ 0 0
$$87$$ −14.5616 −1.56116
$$88$$ 0 0
$$89$$ 7.12311 0.755048 0.377524 0.926000i $$-0.376776\pi$$
0.377524 + 0.926000i $$0.376776\pi$$
$$90$$ 0 0
$$91$$ −4.56155 −0.478181
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −1.12311 −0.115228
$$96$$ 0 0
$$97$$ −14.8078 −1.50350 −0.751750 0.659448i $$-0.770790\pi$$
−0.751750 + 0.659448i $$0.770790\pi$$
$$98$$ 0 0
$$99$$ 9.12311 0.916907
$$100$$ 0 0
$$101$$ −0.246211 −0.0244989 −0.0122495 0.999925i $$-0.503899\pi$$
−0.0122495 + 0.999925i $$0.503899\pi$$
$$102$$ 0 0
$$103$$ −1.43845 −0.141734 −0.0708672 0.997486i $$-0.522577\pi$$
−0.0708672 + 0.997486i $$0.522577\pi$$
$$104$$ 0 0
$$105$$ 2.56155 0.249982
$$106$$ 0 0
$$107$$ −11.3693 −1.09911 −0.549557 0.835456i $$-0.685203\pi$$
−0.549557 + 0.835456i $$0.685203\pi$$
$$108$$ 0 0
$$109$$ −17.6847 −1.69388 −0.846942 0.531686i $$-0.821559\pi$$
−0.846942 + 0.531686i $$0.821559\pi$$
$$110$$ 0 0
$$111$$ 15.3693 1.45879
$$112$$ 0 0
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 0 0
$$115$$ −5.12311 −0.477732
$$116$$ 0 0
$$117$$ −16.2462 −1.50196
$$118$$ 0 0
$$119$$ −4.56155 −0.418157
$$120$$ 0 0
$$121$$ −4.43845 −0.403495
$$122$$ 0 0
$$123$$ 8.00000 0.721336
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −10.2462 −0.909204 −0.454602 0.890695i $$-0.650219\pi$$
−0.454602 + 0.890695i $$0.650219\pi$$
$$128$$ 0 0
$$129$$ −23.3693 −2.05755
$$130$$ 0 0
$$131$$ −9.12311 −0.797089 −0.398545 0.917149i $$-0.630485\pi$$
−0.398545 + 0.917149i $$0.630485\pi$$
$$132$$ 0 0
$$133$$ 1.12311 0.0973856
$$134$$ 0 0
$$135$$ 1.43845 0.123802
$$136$$ 0 0
$$137$$ −8.87689 −0.758404 −0.379202 0.925314i $$-0.623801\pi$$
−0.379202 + 0.925314i $$0.623801\pi$$
$$138$$ 0 0
$$139$$ −6.87689 −0.583291 −0.291645 0.956527i $$-0.594203\pi$$
−0.291645 + 0.956527i $$0.594203\pi$$
$$140$$ 0 0
$$141$$ 9.43845 0.794861
$$142$$ 0 0
$$143$$ −11.6847 −0.977120
$$144$$ 0 0
$$145$$ −5.68466 −0.472085
$$146$$ 0 0
$$147$$ −2.56155 −0.211273
$$148$$ 0 0
$$149$$ 4.24621 0.347863 0.173932 0.984758i $$-0.444353\pi$$
0.173932 + 0.984758i $$0.444353\pi$$
$$150$$ 0 0
$$151$$ −21.9309 −1.78471 −0.892354 0.451335i $$-0.850948\pi$$
−0.892354 + 0.451335i $$0.850948\pi$$
$$152$$ 0 0
$$153$$ −16.2462 −1.31343
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −3.75379 −0.299585 −0.149792 0.988717i $$-0.547861\pi$$
−0.149792 + 0.988717i $$0.547861\pi$$
$$158$$ 0 0
$$159$$ 8.00000 0.634441
$$160$$ 0 0
$$161$$ 5.12311 0.403757
$$162$$ 0 0
$$163$$ 1.12311 0.0879684 0.0439842 0.999032i $$-0.485995\pi$$
0.0439842 + 0.999032i $$0.485995\pi$$
$$164$$ 0 0
$$165$$ 6.56155 0.510816
$$166$$ 0 0
$$167$$ −21.9309 −1.69706 −0.848531 0.529146i $$-0.822512\pi$$
−0.848531 + 0.529146i $$0.822512\pi$$
$$168$$ 0 0
$$169$$ 7.80776 0.600597
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 0 0
$$173$$ 8.56155 0.650923 0.325461 0.945555i $$-0.394480\pi$$
0.325461 + 0.945555i $$0.394480\pi$$
$$174$$ 0 0
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ 10.2462 0.770152
$$178$$ 0 0
$$179$$ 20.0000 1.49487 0.747435 0.664335i $$-0.231285\pi$$
0.747435 + 0.664335i $$0.231285\pi$$
$$180$$ 0 0
$$181$$ −23.6155 −1.75533 −0.877664 0.479276i $$-0.840899\pi$$
−0.877664 + 0.479276i $$0.840899\pi$$
$$182$$ 0 0
$$183$$ −24.0000 −1.77413
$$184$$ 0 0
$$185$$ 6.00000 0.441129
$$186$$ 0 0
$$187$$ −11.6847 −0.854467
$$188$$ 0 0
$$189$$ −1.43845 −0.104632
$$190$$ 0 0
$$191$$ 9.43845 0.682942 0.341471 0.939892i $$-0.389075\pi$$
0.341471 + 0.939892i $$0.389075\pi$$
$$192$$ 0 0
$$193$$ −5.36932 −0.386492 −0.193246 0.981150i $$-0.561902\pi$$
−0.193246 + 0.981150i $$0.561902\pi$$
$$194$$ 0 0
$$195$$ −11.6847 −0.836756
$$196$$ 0 0
$$197$$ 7.12311 0.507500 0.253750 0.967270i $$-0.418336\pi$$
0.253750 + 0.967270i $$0.418336\pi$$
$$198$$ 0 0
$$199$$ 18.2462 1.29344 0.646720 0.762728i $$-0.276140\pi$$
0.646720 + 0.762728i $$0.276140\pi$$
$$200$$ 0 0
$$201$$ 16.0000 1.12855
$$202$$ 0 0
$$203$$ 5.68466 0.398985
$$204$$ 0 0
$$205$$ 3.12311 0.218127
$$206$$ 0 0
$$207$$ 18.2462 1.26820
$$208$$ 0 0
$$209$$ 2.87689 0.198999
$$210$$ 0 0
$$211$$ −23.0540 −1.58710 −0.793551 0.608504i $$-0.791770\pi$$
−0.793551 + 0.608504i $$0.791770\pi$$
$$212$$ 0 0
$$213$$ 20.4924 1.40412
$$214$$ 0 0
$$215$$ −9.12311 −0.622191
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −10.8769 −0.734992
$$220$$ 0 0
$$221$$ 20.8078 1.39968
$$222$$ 0 0
$$223$$ 6.56155 0.439394 0.219697 0.975568i $$-0.429493\pi$$
0.219697 + 0.975568i $$0.429493\pi$$
$$224$$ 0 0
$$225$$ 3.56155 0.237437
$$226$$ 0 0
$$227$$ 23.6847 1.57201 0.786003 0.618223i $$-0.212147\pi$$
0.786003 + 0.618223i $$0.212147\pi$$
$$228$$ 0 0
$$229$$ −19.1231 −1.26369 −0.631845 0.775095i $$-0.717702\pi$$
−0.631845 + 0.775095i $$0.717702\pi$$
$$230$$ 0 0
$$231$$ −6.56155 −0.431718
$$232$$ 0 0
$$233$$ −3.12311 −0.204601 −0.102301 0.994754i $$-0.532620\pi$$
−0.102301 + 0.994754i $$0.532620\pi$$
$$234$$ 0 0
$$235$$ 3.68466 0.240361
$$236$$ 0 0
$$237$$ −16.8078 −1.09178
$$238$$ 0 0
$$239$$ 0.807764 0.0522499 0.0261250 0.999659i $$-0.491683\pi$$
0.0261250 + 0.999659i $$0.491683\pi$$
$$240$$ 0 0
$$241$$ 12.2462 0.788848 0.394424 0.918929i $$-0.370944\pi$$
0.394424 + 0.918929i $$0.370944\pi$$
$$242$$ 0 0
$$243$$ 22.2462 1.42710
$$244$$ 0 0
$$245$$ −1.00000 −0.0638877
$$246$$ 0 0
$$247$$ −5.12311 −0.325975
$$248$$ 0 0
$$249$$ −10.2462 −0.649327
$$250$$ 0 0
$$251$$ −17.1231 −1.08080 −0.540400 0.841408i $$-0.681727\pi$$
−0.540400 + 0.841408i $$0.681727\pi$$
$$252$$ 0 0
$$253$$ 13.1231 0.825043
$$254$$ 0 0
$$255$$ −11.6847 −0.731722
$$256$$ 0 0
$$257$$ 22.4924 1.40304 0.701519 0.712650i $$-0.252505\pi$$
0.701519 + 0.712650i $$0.252505\pi$$
$$258$$ 0 0
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ 20.2462 1.25321
$$262$$ 0 0
$$263$$ 21.1231 1.30251 0.651253 0.758860i $$-0.274244\pi$$
0.651253 + 0.758860i $$0.274244\pi$$
$$264$$ 0 0
$$265$$ 3.12311 0.191851
$$266$$ 0 0
$$267$$ −18.2462 −1.11665
$$268$$ 0 0
$$269$$ −28.7386 −1.75223 −0.876113 0.482106i $$-0.839872\pi$$
−0.876113 + 0.482106i $$0.839872\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 0 0
$$273$$ 11.6847 0.707188
$$274$$ 0 0
$$275$$ 2.56155 0.154467
$$276$$ 0 0
$$277$$ −16.2462 −0.976140 −0.488070 0.872804i $$-0.662299\pi$$
−0.488070 + 0.872804i $$0.662299\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 16.5616 0.987979 0.493990 0.869468i $$-0.335538\pi$$
0.493990 + 0.869468i $$0.335538\pi$$
$$282$$ 0 0
$$283$$ −23.6847 −1.40791 −0.703953 0.710246i $$-0.748584\pi$$
−0.703953 + 0.710246i $$0.748584\pi$$
$$284$$ 0 0
$$285$$ 2.87689 0.170413
$$286$$ 0 0
$$287$$ −3.12311 −0.184351
$$288$$ 0 0
$$289$$ 3.80776 0.223986
$$290$$ 0 0
$$291$$ 37.9309 2.22355
$$292$$ 0 0
$$293$$ −9.68466 −0.565784 −0.282892 0.959152i $$-0.591294\pi$$
−0.282892 + 0.959152i $$0.591294\pi$$
$$294$$ 0 0
$$295$$ 4.00000 0.232889
$$296$$ 0 0
$$297$$ −3.68466 −0.213806
$$298$$ 0 0
$$299$$ −23.3693 −1.35148
$$300$$ 0 0
$$301$$ 9.12311 0.525847
$$302$$ 0 0
$$303$$ 0.630683 0.0362318
$$304$$ 0 0
$$305$$ −9.36932 −0.536486
$$306$$ 0 0
$$307$$ −31.6847 −1.80834 −0.904169 0.427174i $$-0.859509\pi$$
−0.904169 + 0.427174i $$0.859509\pi$$
$$308$$ 0 0
$$309$$ 3.68466 0.209613
$$310$$ 0 0
$$311$$ 9.61553 0.545247 0.272623 0.962121i $$-0.412109\pi$$
0.272623 + 0.962121i $$0.412109\pi$$
$$312$$ 0 0
$$313$$ 31.3002 1.76919 0.884596 0.466359i $$-0.154434\pi$$
0.884596 + 0.466359i $$0.154434\pi$$
$$314$$ 0 0
$$315$$ −3.56155 −0.200671
$$316$$ 0 0
$$317$$ 22.4924 1.26330 0.631650 0.775254i $$-0.282378\pi$$
0.631650 + 0.775254i $$0.282378\pi$$
$$318$$ 0 0
$$319$$ 14.5616 0.815290
$$320$$ 0 0
$$321$$ 29.1231 1.62549
$$322$$ 0 0
$$323$$ −5.12311 −0.285057
$$324$$ 0 0
$$325$$ −4.56155 −0.253029
$$326$$ 0 0
$$327$$ 45.3002 2.50511
$$328$$ 0 0
$$329$$ −3.68466 −0.203142
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ 0 0
$$333$$ −21.3693 −1.17103
$$334$$ 0 0
$$335$$ 6.24621 0.341267
$$336$$ 0 0
$$337$$ −34.4924 −1.87892 −0.939461 0.342656i $$-0.888674\pi$$
−0.939461 + 0.342656i $$0.888674\pi$$
$$338$$ 0 0
$$339$$ 35.8617 1.94774
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 13.1231 0.706524
$$346$$ 0 0
$$347$$ −1.12311 −0.0602915 −0.0301457 0.999546i $$-0.509597\pi$$
−0.0301457 + 0.999546i $$0.509597\pi$$
$$348$$ 0 0
$$349$$ 22.4924 1.20399 0.601996 0.798499i $$-0.294372\pi$$
0.601996 + 0.798499i $$0.294372\pi$$
$$350$$ 0 0
$$351$$ 6.56155 0.350230
$$352$$ 0 0
$$353$$ −14.8078 −0.788138 −0.394069 0.919081i $$-0.628933\pi$$
−0.394069 + 0.919081i $$0.628933\pi$$
$$354$$ 0 0
$$355$$ 8.00000 0.424596
$$356$$ 0 0
$$357$$ 11.6847 0.618418
$$358$$ 0 0
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ 0 0
$$361$$ −17.7386 −0.933612
$$362$$ 0 0
$$363$$ 11.3693 0.596734
$$364$$ 0 0
$$365$$ −4.24621 −0.222257
$$366$$ 0 0
$$367$$ −3.68466 −0.192338 −0.0961688 0.995365i $$-0.530659\pi$$
−0.0961688 + 0.995365i $$0.530659\pi$$
$$368$$ 0 0
$$369$$ −11.1231 −0.579046
$$370$$ 0 0
$$371$$ −3.12311 −0.162144
$$372$$ 0 0
$$373$$ −29.3693 −1.52069 −0.760343 0.649522i $$-0.774969\pi$$
−0.760343 + 0.649522i $$0.774969\pi$$
$$374$$ 0 0
$$375$$ 2.56155 0.132278
$$376$$ 0 0
$$377$$ −25.9309 −1.33551
$$378$$ 0 0
$$379$$ 16.4924 0.847159 0.423579 0.905859i $$-0.360773\pi$$
0.423579 + 0.905859i $$0.360773\pi$$
$$380$$ 0 0
$$381$$ 26.2462 1.34463
$$382$$ 0 0
$$383$$ 10.2462 0.523557 0.261778 0.965128i $$-0.415691\pi$$
0.261778 + 0.965128i $$0.415691\pi$$
$$384$$ 0 0
$$385$$ −2.56155 −0.130549
$$386$$ 0 0
$$387$$ 32.4924 1.65168
$$388$$ 0 0
$$389$$ −3.93087 −0.199303 −0.0996515 0.995022i $$-0.531773\pi$$
−0.0996515 + 0.995022i $$0.531773\pi$$
$$390$$ 0 0
$$391$$ −23.3693 −1.18184
$$392$$ 0 0
$$393$$ 23.3693 1.17883
$$394$$ 0 0
$$395$$ −6.56155 −0.330148
$$396$$ 0 0
$$397$$ −23.4384 −1.17634 −0.588171 0.808737i $$-0.700152\pi$$
−0.588171 + 0.808737i $$0.700152\pi$$
$$398$$ 0 0
$$399$$ −2.87689 −0.144025
$$400$$ 0 0
$$401$$ 27.4384 1.37021 0.685105 0.728444i $$-0.259756\pi$$
0.685105 + 0.728444i $$0.259756\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 7.00000 0.347833
$$406$$ 0 0
$$407$$ −15.3693 −0.761829
$$408$$ 0 0
$$409$$ −26.4924 −1.30997 −0.654983 0.755644i $$-0.727324\pi$$
−0.654983 + 0.755644i $$0.727324\pi$$
$$410$$ 0 0
$$411$$ 22.7386 1.12161
$$412$$ 0 0
$$413$$ −4.00000 −0.196827
$$414$$ 0 0
$$415$$ −4.00000 −0.196352
$$416$$ 0 0
$$417$$ 17.6155 0.862636
$$418$$ 0 0
$$419$$ 9.75379 0.476504 0.238252 0.971203i $$-0.423426\pi$$
0.238252 + 0.971203i $$0.423426\pi$$
$$420$$ 0 0
$$421$$ −9.68466 −0.472001 −0.236001 0.971753i $$-0.575837\pi$$
−0.236001 + 0.971753i $$0.575837\pi$$
$$422$$ 0 0
$$423$$ −13.1231 −0.638067
$$424$$ 0 0
$$425$$ −4.56155 −0.221268
$$426$$ 0 0
$$427$$ 9.36932 0.453413
$$428$$ 0 0
$$429$$ 29.9309 1.44508
$$430$$ 0 0
$$431$$ −0.807764 −0.0389086 −0.0194543 0.999811i $$-0.506193\pi$$
−0.0194543 + 0.999811i $$0.506193\pi$$
$$432$$ 0 0
$$433$$ −8.24621 −0.396288 −0.198144 0.980173i $$-0.563491\pi$$
−0.198144 + 0.980173i $$0.563491\pi$$
$$434$$ 0 0
$$435$$ 14.5616 0.698173
$$436$$ 0 0
$$437$$ 5.75379 0.275241
$$438$$ 0 0
$$439$$ 15.3693 0.733537 0.366769 0.930312i $$-0.380464\pi$$
0.366769 + 0.930312i $$0.380464\pi$$
$$440$$ 0 0
$$441$$ 3.56155 0.169598
$$442$$ 0 0
$$443$$ −27.3693 −1.30036 −0.650178 0.759782i $$-0.725306\pi$$
−0.650178 + 0.759782i $$0.725306\pi$$
$$444$$ 0 0
$$445$$ −7.12311 −0.337668
$$446$$ 0 0
$$447$$ −10.8769 −0.514459
$$448$$ 0 0
$$449$$ 18.8078 0.887593 0.443797 0.896128i $$-0.353631\pi$$
0.443797 + 0.896128i $$0.353631\pi$$
$$450$$ 0 0
$$451$$ −8.00000 −0.376705
$$452$$ 0 0
$$453$$ 56.1771 2.63943
$$454$$ 0 0
$$455$$ 4.56155 0.213849
$$456$$ 0 0
$$457$$ −8.87689 −0.415244 −0.207622 0.978209i $$-0.566572\pi$$
−0.207622 + 0.978209i $$0.566572\pi$$
$$458$$ 0 0
$$459$$ 6.56155 0.306267
$$460$$ 0 0
$$461$$ 4.87689 0.227140 0.113570 0.993530i $$-0.463771\pi$$
0.113570 + 0.993530i $$0.463771\pi$$
$$462$$ 0 0
$$463$$ 20.4924 0.952364 0.476182 0.879347i $$-0.342020\pi$$
0.476182 + 0.879347i $$0.342020\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 26.5616 1.22912 0.614561 0.788869i $$-0.289333\pi$$
0.614561 + 0.788869i $$0.289333\pi$$
$$468$$ 0 0
$$469$$ −6.24621 −0.288423
$$470$$ 0 0
$$471$$ 9.61553 0.443060
$$472$$ 0 0
$$473$$ 23.3693 1.07452
$$474$$ 0 0
$$475$$ 1.12311 0.0515316
$$476$$ 0 0
$$477$$ −11.1231 −0.509292
$$478$$ 0 0
$$479$$ −13.1231 −0.599610 −0.299805 0.954001i $$-0.596922\pi$$
−0.299805 + 0.954001i $$0.596922\pi$$
$$480$$ 0 0
$$481$$ 27.3693 1.24793
$$482$$ 0 0
$$483$$ −13.1231 −0.597122
$$484$$ 0 0
$$485$$ 14.8078 0.672386
$$486$$ 0 0
$$487$$ −5.12311 −0.232150 −0.116075 0.993240i $$-0.537031\pi$$
−0.116075 + 0.993240i $$0.537031\pi$$
$$488$$ 0 0
$$489$$ −2.87689 −0.130098
$$490$$ 0 0
$$491$$ 4.17708 0.188509 0.0942545 0.995548i $$-0.469953\pi$$
0.0942545 + 0.995548i $$0.469953\pi$$
$$492$$ 0 0
$$493$$ −25.9309 −1.16787
$$494$$ 0 0
$$495$$ −9.12311 −0.410053
$$496$$ 0 0
$$497$$ −8.00000 −0.358849
$$498$$ 0 0
$$499$$ −4.17708 −0.186992 −0.0934959 0.995620i $$-0.529804\pi$$
−0.0934959 + 0.995620i $$0.529804\pi$$
$$500$$ 0 0
$$501$$ 56.1771 2.50981
$$502$$ 0 0
$$503$$ −10.0691 −0.448960 −0.224480 0.974479i $$-0.572068\pi$$
−0.224480 + 0.974479i $$0.572068\pi$$
$$504$$ 0 0
$$505$$ 0.246211 0.0109563
$$506$$ 0 0
$$507$$ −20.0000 −0.888231
$$508$$ 0 0
$$509$$ 28.2462 1.25199 0.625996 0.779827i $$-0.284693\pi$$
0.625996 + 0.779827i $$0.284693\pi$$
$$510$$ 0 0
$$511$$ 4.24621 0.187841
$$512$$ 0 0
$$513$$ −1.61553 −0.0713273
$$514$$ 0 0
$$515$$ 1.43845 0.0633856
$$516$$ 0 0
$$517$$ −9.43845 −0.415102
$$518$$ 0 0
$$519$$ −21.9309 −0.962658
$$520$$ 0 0
$$521$$ 10.0000 0.438108 0.219054 0.975713i $$-0.429703\pi$$
0.219054 + 0.975713i $$0.429703\pi$$
$$522$$ 0 0
$$523$$ 7.50758 0.328283 0.164142 0.986437i $$-0.447515\pi$$
0.164142 + 0.986437i $$0.447515\pi$$
$$524$$ 0 0
$$525$$ −2.56155 −0.111795
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 3.24621 0.141140
$$530$$ 0 0
$$531$$ −14.2462 −0.618233
$$532$$ 0 0
$$533$$ 14.2462 0.617072
$$534$$ 0 0
$$535$$ 11.3693 0.491538
$$536$$ 0 0
$$537$$ −51.2311 −2.21078
$$538$$ 0 0
$$539$$ 2.56155 0.110334
$$540$$ 0 0
$$541$$ 17.1922 0.739152 0.369576 0.929201i $$-0.379503\pi$$
0.369576 + 0.929201i $$0.379503\pi$$
$$542$$ 0 0
$$543$$ 60.4924 2.59598
$$544$$ 0 0
$$545$$ 17.6847 0.757528
$$546$$ 0 0
$$547$$ 14.2462 0.609124 0.304562 0.952493i $$-0.401490\pi$$
0.304562 + 0.952493i $$0.401490\pi$$
$$548$$ 0 0
$$549$$ 33.3693 1.42417
$$550$$ 0 0
$$551$$ 6.38447 0.271988
$$552$$ 0 0
$$553$$ 6.56155 0.279026
$$554$$ 0 0
$$555$$ −15.3693 −0.652391
$$556$$ 0 0
$$557$$ 4.87689 0.206641 0.103320 0.994648i $$-0.467053\pi$$
0.103320 + 0.994648i $$0.467053\pi$$
$$558$$ 0 0
$$559$$ −41.6155 −1.76015
$$560$$ 0 0
$$561$$ 29.9309 1.26368
$$562$$ 0 0
$$563$$ −28.0000 −1.18006 −0.590030 0.807382i $$-0.700884\pi$$
−0.590030 + 0.807382i $$0.700884\pi$$
$$564$$ 0 0
$$565$$ 14.0000 0.588984
$$566$$ 0 0
$$567$$ −7.00000 −0.293972
$$568$$ 0 0
$$569$$ 34.9848 1.46664 0.733320 0.679883i $$-0.237969\pi$$
0.733320 + 0.679883i $$0.237969\pi$$
$$570$$ 0 0
$$571$$ 7.50758 0.314182 0.157091 0.987584i $$-0.449788\pi$$
0.157091 + 0.987584i $$0.449788\pi$$
$$572$$ 0 0
$$573$$ −24.1771 −1.01001
$$574$$ 0 0
$$575$$ 5.12311 0.213648
$$576$$ 0 0
$$577$$ 13.0540 0.543444 0.271722 0.962376i $$-0.412407\pi$$
0.271722 + 0.962376i $$0.412407\pi$$
$$578$$ 0 0
$$579$$ 13.7538 0.571588
$$580$$ 0 0
$$581$$ 4.00000 0.165948
$$582$$ 0 0
$$583$$ −8.00000 −0.331326
$$584$$ 0 0
$$585$$ 16.2462 0.671698
$$586$$ 0 0
$$587$$ −9.75379 −0.402582 −0.201291 0.979531i $$-0.564514\pi$$
−0.201291 + 0.979531i $$0.564514\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −18.2462 −0.750549
$$592$$ 0 0
$$593$$ −23.4384 −0.962502 −0.481251 0.876583i $$-0.659817\pi$$
−0.481251 + 0.876583i $$0.659817\pi$$
$$594$$ 0 0
$$595$$ 4.56155 0.187005
$$596$$ 0 0
$$597$$ −46.7386 −1.91288
$$598$$ 0 0
$$599$$ −8.80776 −0.359875 −0.179938 0.983678i $$-0.557590\pi$$
−0.179938 + 0.983678i $$0.557590\pi$$
$$600$$ 0 0
$$601$$ −26.4924 −1.08065 −0.540324 0.841457i $$-0.681698\pi$$
−0.540324 + 0.841457i $$0.681698\pi$$
$$602$$ 0 0
$$603$$ −22.2462 −0.905936
$$604$$ 0 0
$$605$$ 4.43845 0.180449
$$606$$ 0 0
$$607$$ 4.94602 0.200753 0.100376 0.994950i $$-0.467995\pi$$
0.100376 + 0.994950i $$0.467995\pi$$
$$608$$ 0 0
$$609$$ −14.5616 −0.590064
$$610$$ 0 0
$$611$$ 16.8078 0.679969
$$612$$ 0 0
$$613$$ 8.73863 0.352950 0.176475 0.984305i $$-0.443531\pi$$
0.176475 + 0.984305i $$0.443531\pi$$
$$614$$ 0 0
$$615$$ −8.00000 −0.322591
$$616$$ 0 0
$$617$$ 15.7538 0.634224 0.317112 0.948388i $$-0.397287\pi$$
0.317112 + 0.948388i $$0.397287\pi$$
$$618$$ 0 0
$$619$$ −42.1080 −1.69246 −0.846231 0.532817i $$-0.821134\pi$$
−0.846231 + 0.532817i $$0.821134\pi$$
$$620$$ 0 0
$$621$$ −7.36932 −0.295720
$$622$$ 0 0
$$623$$ 7.12311 0.285381
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −7.36932 −0.294302
$$628$$ 0 0
$$629$$ 27.3693 1.09129
$$630$$ 0 0
$$631$$ −8.80776 −0.350632 −0.175316 0.984512i $$-0.556095\pi$$
−0.175316 + 0.984512i $$0.556095\pi$$
$$632$$ 0 0
$$633$$ 59.0540 2.34718
$$634$$ 0 0
$$635$$ 10.2462 0.406608
$$636$$ 0 0
$$637$$ −4.56155 −0.180735
$$638$$ 0 0
$$639$$ −28.4924 −1.12714
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 0 0
$$643$$ −2.56155 −0.101018 −0.0505089 0.998724i $$-0.516084\pi$$
−0.0505089 + 0.998724i $$0.516084\pi$$
$$644$$ 0 0
$$645$$ 23.3693 0.920166
$$646$$ 0 0
$$647$$ −3.50758 −0.137897 −0.0689486 0.997620i $$-0.521964\pi$$
−0.0689486 + 0.997620i $$0.521964\pi$$
$$648$$ 0 0
$$649$$ −10.2462 −0.402199
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −49.2311 −1.92656 −0.963280 0.268499i $$-0.913473\pi$$
−0.963280 + 0.268499i $$0.913473\pi$$
$$654$$ 0 0
$$655$$ 9.12311 0.356469
$$656$$ 0 0
$$657$$ 15.1231 0.590009
$$658$$ 0 0
$$659$$ −36.1771 −1.40926 −0.704629 0.709575i $$-0.748887\pi$$
−0.704629 + 0.709575i $$0.748887\pi$$
$$660$$ 0 0
$$661$$ −3.12311 −0.121475 −0.0607374 0.998154i $$-0.519345\pi$$
−0.0607374 + 0.998154i $$0.519345\pi$$
$$662$$ 0 0
$$663$$ −53.3002 −2.07001
$$664$$ 0 0
$$665$$ −1.12311 −0.0435522
$$666$$ 0 0
$$667$$ 29.1231 1.12765
$$668$$ 0 0
$$669$$ −16.8078 −0.649826
$$670$$ 0 0
$$671$$ 24.0000 0.926510
$$672$$ 0 0
$$673$$ −25.8617 −0.996897 −0.498448 0.866919i $$-0.666097\pi$$
−0.498448 + 0.866919i $$0.666097\pi$$
$$674$$ 0 0
$$675$$ −1.43845 −0.0553659
$$676$$ 0 0
$$677$$ 23.9309 0.919738 0.459869 0.887987i $$-0.347896\pi$$
0.459869 + 0.887987i $$0.347896\pi$$
$$678$$ 0 0
$$679$$ −14.8078 −0.568270
$$680$$ 0 0
$$681$$ −60.6695 −2.32486
$$682$$ 0 0
$$683$$ 42.7386 1.63535 0.817674 0.575681i $$-0.195263\pi$$
0.817674 + 0.575681i $$0.195263\pi$$
$$684$$ 0 0
$$685$$ 8.87689 0.339169
$$686$$ 0 0
$$687$$ 48.9848 1.86889
$$688$$ 0 0
$$689$$ 14.2462 0.542737
$$690$$ 0 0
$$691$$ 8.49242 0.323067 0.161533 0.986867i $$-0.448356\pi$$
0.161533 + 0.986867i $$0.448356\pi$$
$$692$$ 0 0
$$693$$ 9.12311 0.346558
$$694$$ 0 0
$$695$$ 6.87689 0.260855
$$696$$ 0 0
$$697$$ 14.2462 0.539614
$$698$$ 0 0
$$699$$ 8.00000 0.302588
$$700$$ 0 0
$$701$$ −0.0691303 −0.00261102 −0.00130551 0.999999i $$-0.500416\pi$$
−0.00130551 + 0.999999i $$0.500416\pi$$
$$702$$ 0 0
$$703$$ −6.73863 −0.254152
$$704$$ 0 0
$$705$$ −9.43845 −0.355472
$$706$$ 0 0
$$707$$ −0.246211 −0.00925973
$$708$$ 0 0
$$709$$ 18.1771 0.682655 0.341327 0.939945i $$-0.389124\pi$$
0.341327 + 0.939945i $$0.389124\pi$$
$$710$$ 0 0
$$711$$ 23.3693 0.876418
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 11.6847 0.436981
$$716$$ 0 0
$$717$$ −2.06913 −0.0772731
$$718$$ 0 0
$$719$$ −49.6155 −1.85035 −0.925173 0.379544i $$-0.876081\pi$$
−0.925173 + 0.379544i $$0.876081\pi$$
$$720$$ 0 0
$$721$$ −1.43845 −0.0535706
$$722$$ 0 0
$$723$$ −31.3693 −1.16664
$$724$$ 0 0
$$725$$ 5.68466 0.211123
$$726$$ 0 0
$$727$$ −19.5076 −0.723496 −0.361748 0.932276i $$-0.617820\pi$$
−0.361748 + 0.932276i $$0.617820\pi$$
$$728$$ 0 0
$$729$$ −35.9848 −1.33277
$$730$$ 0 0
$$731$$ −41.6155 −1.53921
$$732$$ 0 0
$$733$$ 5.68466 0.209968 0.104984 0.994474i $$-0.466521\pi$$
0.104984 + 0.994474i $$0.466521\pi$$
$$734$$ 0 0
$$735$$ 2.56155 0.0944843
$$736$$ 0 0
$$737$$ −16.0000 −0.589368
$$738$$ 0 0
$$739$$ 6.06913 0.223257 0.111628 0.993750i $$-0.464393\pi$$
0.111628 + 0.993750i $$0.464393\pi$$
$$740$$ 0 0
$$741$$ 13.1231 0.482089
$$742$$ 0 0
$$743$$ −32.9848 −1.21010 −0.605048 0.796189i $$-0.706846\pi$$
−0.605048 + 0.796189i $$0.706846\pi$$
$$744$$ 0 0
$$745$$ −4.24621 −0.155569
$$746$$ 0 0
$$747$$ 14.2462 0.521242
$$748$$ 0 0
$$749$$ −11.3693 −0.415426
$$750$$ 0 0
$$751$$ −45.9309 −1.67604 −0.838021 0.545639i $$-0.816287\pi$$
−0.838021 + 0.545639i $$0.816287\pi$$
$$752$$ 0 0
$$753$$ 43.8617 1.59841
$$754$$ 0 0
$$755$$ 21.9309 0.798146
$$756$$ 0 0
$$757$$ −14.6307 −0.531761 −0.265881 0.964006i $$-0.585663\pi$$
−0.265881 + 0.964006i $$0.585663\pi$$
$$758$$ 0 0
$$759$$ −33.6155 −1.22017
$$760$$ 0 0
$$761$$ 31.7538 1.15107 0.575537 0.817776i $$-0.304793\pi$$
0.575537 + 0.817776i $$0.304793\pi$$
$$762$$ 0 0
$$763$$ −17.6847 −0.640228
$$764$$ 0 0
$$765$$ 16.2462 0.587383
$$766$$ 0 0
$$767$$ 18.2462 0.658833
$$768$$ 0 0
$$769$$ −9.50758 −0.342852 −0.171426 0.985197i $$-0.554837\pi$$
−0.171426 + 0.985197i $$0.554837\pi$$
$$770$$ 0 0
$$771$$ −57.6155 −2.07497
$$772$$ 0 0
$$773$$ −8.06913 −0.290226 −0.145113 0.989415i $$-0.546355\pi$$
−0.145113 + 0.989415i $$0.546355\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 15.3693 0.551371
$$778$$ 0 0
$$779$$ −3.50758 −0.125672
$$780$$ 0 0
$$781$$ −20.4924 −0.733277
$$782$$ 0 0
$$783$$ −8.17708 −0.292225
$$784$$ 0 0
$$785$$ 3.75379 0.133978
$$786$$ 0 0
$$787$$ −3.82292 −0.136272 −0.0681362 0.997676i $$-0.521705\pi$$
−0.0681362 + 0.997676i $$0.521705\pi$$
$$788$$ 0 0
$$789$$ −54.1080 −1.92629
$$790$$ 0 0
$$791$$ −14.0000 −0.497783
$$792$$ 0 0
$$793$$ −42.7386 −1.51769
$$794$$ 0 0
$$795$$ −8.00000 −0.283731
$$796$$ 0 0
$$797$$ 13.0540 0.462396 0.231198 0.972907i $$-0.425736\pi$$
0.231198 + 0.972907i $$0.425736\pi$$
$$798$$ 0 0
$$799$$ 16.8078 0.594616
$$800$$ 0 0
$$801$$ 25.3693 0.896381
$$802$$ 0 0
$$803$$ 10.8769 0.383837
$$804$$ 0 0
$$805$$ −5.12311 −0.180566
$$806$$ 0 0
$$807$$ 73.6155 2.59139
$$808$$ 0 0
$$809$$ −53.5464 −1.88259 −0.941296 0.337584i $$-0.890390\pi$$
−0.941296 + 0.337584i $$0.890390\pi$$
$$810$$ 0 0
$$811$$ −21.6155 −0.759024 −0.379512 0.925187i $$-0.623908\pi$$
−0.379512 + 0.925187i $$0.623908\pi$$
$$812$$ 0 0
$$813$$ −40.9848 −1.43740
$$814$$ 0 0
$$815$$ −1.12311 −0.0393407
$$816$$ 0 0
$$817$$ 10.2462 0.358470
$$818$$ 0 0
$$819$$ −16.2462 −0.567689
$$820$$ 0 0
$$821$$ −40.4233 −1.41078 −0.705391 0.708818i $$-0.749229\pi$$
−0.705391 + 0.708818i $$0.749229\pi$$
$$822$$ 0 0
$$823$$ 3.50758 0.122266 0.0611332 0.998130i $$-0.480529\pi$$
0.0611332 + 0.998130i $$0.480529\pi$$
$$824$$ 0 0
$$825$$ −6.56155 −0.228444
$$826$$ 0 0
$$827$$ 19.3693 0.673537 0.336769 0.941587i $$-0.390666\pi$$
0.336769 + 0.941587i $$0.390666\pi$$
$$828$$ 0 0
$$829$$ −43.1231 −1.49773 −0.748864 0.662724i $$-0.769400\pi$$
−0.748864 + 0.662724i $$0.769400\pi$$
$$830$$ 0 0
$$831$$ 41.6155 1.44363
$$832$$ 0 0
$$833$$ −4.56155 −0.158048
$$834$$ 0 0
$$835$$ 21.9309 0.758949
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 37.1231 1.28163 0.640816 0.767695i $$-0.278596\pi$$
0.640816 + 0.767695i $$0.278596\pi$$
$$840$$ 0 0
$$841$$ 3.31534 0.114322
$$842$$ 0 0
$$843$$ −42.4233 −1.46114
$$844$$ 0 0
$$845$$ −7.80776 −0.268595
$$846$$ 0 0
$$847$$ −4.43845 −0.152507
$$848$$ 0 0
$$849$$ 60.6695 2.08217
$$850$$ 0 0
$$851$$ −30.7386 −1.05371
$$852$$ 0 0
$$853$$ 56.7386 1.94269 0.971347 0.237666i $$-0.0763824\pi$$
0.971347 + 0.237666i $$0.0763824\pi$$
$$854$$ 0 0
$$855$$ −4.00000 −0.136797
$$856$$ 0 0
$$857$$ −32.2462 −1.10151 −0.550755 0.834667i $$-0.685660\pi$$
−0.550755 + 0.834667i $$0.685660\pi$$
$$858$$ 0 0
$$859$$ 16.4924 0.562714 0.281357 0.959603i $$-0.409215\pi$$
0.281357 + 0.959603i $$0.409215\pi$$
$$860$$ 0 0
$$861$$ 8.00000 0.272639
$$862$$ 0 0
$$863$$ 42.2462 1.43808 0.719039 0.694970i $$-0.244582\pi$$
0.719039 + 0.694970i $$0.244582\pi$$
$$864$$ 0 0
$$865$$ −8.56155 −0.291102
$$866$$ 0 0
$$867$$ −9.75379 −0.331256
$$868$$ 0 0
$$869$$ 16.8078 0.570164
$$870$$ 0 0
$$871$$ 28.4924 0.965429
$$872$$ 0 0
$$873$$ −52.7386 −1.78493
$$874$$ 0 0
$$875$$ −1.00000 −0.0338062
$$876$$ 0 0
$$877$$ 23.7538 0.802108 0.401054 0.916054i $$-0.368644\pi$$
0.401054 + 0.916054i $$0.368644\pi$$
$$878$$ 0 0
$$879$$ 24.8078 0.836745
$$880$$ 0 0
$$881$$ 45.8617 1.54512 0.772561 0.634941i $$-0.218976\pi$$
0.772561 + 0.634941i $$0.218976\pi$$
$$882$$ 0 0
$$883$$ 24.4924 0.824236 0.412118 0.911131i $$-0.364789\pi$$
0.412118 + 0.911131i $$0.364789\pi$$
$$884$$ 0 0
$$885$$ −10.2462 −0.344423
$$886$$ 0 0
$$887$$ 12.4924 0.419454 0.209727 0.977760i $$-0.432742\pi$$
0.209727 + 0.977760i $$0.432742\pi$$
$$888$$ 0 0
$$889$$ −10.2462 −0.343647
$$890$$ 0 0
$$891$$ −17.9309 −0.600707
$$892$$ 0 0
$$893$$ −4.13826 −0.138482
$$894$$ 0 0
$$895$$ −20.0000 −0.668526
$$896$$ 0 0
$$897$$ 59.8617 1.99873
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 14.2462 0.474610
$$902$$ 0 0
$$903$$ −23.3693 −0.777682
$$904$$ 0 0
$$905$$ 23.6155 0.785007
$$906$$ 0 0
$$907$$ 50.1080 1.66381 0.831904 0.554920i $$-0.187251\pi$$
0.831904 + 0.554920i $$0.187251\pi$$
$$908$$ 0 0
$$909$$ −0.876894 −0.0290848
$$910$$ 0 0
$$911$$ −4.49242 −0.148841 −0.0744203 0.997227i $$-0.523711\pi$$
−0.0744203 + 0.997227i $$0.523711\pi$$
$$912$$ 0 0
$$913$$ 10.2462 0.339100
$$914$$ 0 0
$$915$$ 24.0000 0.793416
$$916$$ 0 0
$$917$$ −9.12311 −0.301271
$$918$$ 0 0
$$919$$ 13.3002 0.438733 0.219366 0.975643i $$-0.429601\pi$$
0.219366 + 0.975643i $$0.429601\pi$$
$$920$$ 0 0
$$921$$ 81.1619 2.67438
$$922$$ 0 0
$$923$$ 36.4924 1.20116
$$924$$ 0 0
$$925$$ −6.00000 −0.197279
$$926$$ 0 0
$$927$$ −5.12311 −0.168265
$$928$$ 0 0
$$929$$ −52.1080 −1.70961 −0.854803 0.518952i $$-0.826322\pi$$
−0.854803 + 0.518952i $$0.826322\pi$$
$$930$$ 0 0
$$931$$ 1.12311 0.0368083
$$932$$ 0 0
$$933$$ −24.6307 −0.806372
$$934$$ 0 0
$$935$$ 11.6847 0.382129
$$936$$ 0 0
$$937$$ 22.6695 0.740580 0.370290 0.928916i $$-0.379258\pi$$
0.370290 + 0.928916i $$0.379258\pi$$
$$938$$ 0 0
$$939$$ −80.1771 −2.61648
$$940$$ 0 0
$$941$$ 13.8617 0.451880 0.225940 0.974141i $$-0.427455\pi$$
0.225940 + 0.974141i $$0.427455\pi$$
$$942$$ 0 0
$$943$$ −16.0000 −0.521032
$$944$$ 0 0
$$945$$ 1.43845 0.0467927
$$946$$ 0 0
$$947$$ 4.00000 0.129983 0.0649913 0.997886i $$-0.479298\pi$$
0.0649913 + 0.997886i $$0.479298\pi$$
$$948$$ 0 0
$$949$$ −19.3693 −0.628755
$$950$$ 0 0
$$951$$ −57.6155 −1.86831
$$952$$ 0 0
$$953$$ −24.8769 −0.805842 −0.402921 0.915235i $$-0.632005\pi$$
−0.402921 + 0.915235i $$0.632005\pi$$
$$954$$ 0 0
$$955$$ −9.43845 −0.305421
$$956$$ 0 0
$$957$$ −37.3002 −1.20574
$$958$$ 0 0
$$959$$ −8.87689 −0.286650
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ −40.4924 −1.30485
$$964$$ 0 0
$$965$$ 5.36932 0.172844
$$966$$ 0 0
$$967$$ 26.8769 0.864303 0.432151 0.901801i $$-0.357755\pi$$
0.432151 + 0.901801i $$0.357755\pi$$
$$968$$ 0 0
$$969$$ 13.1231 0.421575
$$970$$ 0 0
$$971$$ −49.4773 −1.58780 −0.793901 0.608048i $$-0.791953\pi$$
−0.793901 + 0.608048i $$0.791953\pi$$
$$972$$ 0 0
$$973$$ −6.87689 −0.220463
$$974$$ 0 0
$$975$$ 11.6847 0.374209
$$976$$ 0 0
$$977$$ −49.2311 −1.57504 −0.787521 0.616288i $$-0.788636\pi$$
−0.787521 + 0.616288i $$0.788636\pi$$
$$978$$ 0 0
$$979$$ 18.2462 0.583151
$$980$$ 0 0
$$981$$ −62.9848 −2.01095
$$982$$ 0 0
$$983$$ 10.4233 0.332451 0.166226 0.986088i $$-0.446842\pi$$
0.166226 + 0.986088i $$0.446842\pi$$
$$984$$ 0 0
$$985$$ −7.12311 −0.226961
$$986$$ 0 0
$$987$$ 9.43845 0.300429
$$988$$ 0 0
$$989$$ 46.7386 1.48620
$$990$$ 0 0
$$991$$ 20.4924 0.650963 0.325482 0.945548i $$-0.394474\pi$$
0.325482 + 0.945548i $$0.394474\pi$$
$$992$$ 0 0
$$993$$ −30.7386 −0.975461
$$994$$ 0 0
$$995$$ −18.2462 −0.578444
$$996$$ 0 0
$$997$$ −9.68466 −0.306716 −0.153358 0.988171i $$-0.549009\pi$$
−0.153358 + 0.988171i $$0.549009\pi$$
$$998$$ 0 0
$$999$$ 8.63068 0.273063
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 2240.2.a.bd.1.1 2
4.3 odd 2 2240.2.a.bh.1.2 2
8.3 odd 2 35.2.a.b.1.2 2
8.5 even 2 560.2.a.i.1.2 2
24.5 odd 2 5040.2.a.bt.1.2 2
24.11 even 2 315.2.a.e.1.1 2
40.3 even 4 175.2.b.b.99.2 4
40.13 odd 4 2800.2.g.t.449.4 4
40.19 odd 2 175.2.a.f.1.1 2
40.27 even 4 175.2.b.b.99.3 4
40.29 even 2 2800.2.a.bi.1.1 2
40.37 odd 4 2800.2.g.t.449.1 4
56.3 even 6 245.2.e.h.226.1 4
56.11 odd 6 245.2.e.i.226.1 4
56.13 odd 2 3920.2.a.bs.1.1 2
56.19 even 6 245.2.e.h.116.1 4
56.27 even 2 245.2.a.d.1.2 2
56.51 odd 6 245.2.e.i.116.1 4
88.43 even 2 4235.2.a.m.1.1 2
104.51 odd 2 5915.2.a.l.1.1 2
120.59 even 2 1575.2.a.p.1.2 2
120.83 odd 4 1575.2.d.e.1324.3 4
120.107 odd 4 1575.2.d.e.1324.2 4
168.83 odd 2 2205.2.a.x.1.1 2
280.27 odd 4 1225.2.b.f.99.3 4
280.83 odd 4 1225.2.b.f.99.2 4
280.139 even 2 1225.2.a.s.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.2 2 8.3 odd 2
175.2.a.f.1.1 2 40.19 odd 2
175.2.b.b.99.2 4 40.3 even 4
175.2.b.b.99.3 4 40.27 even 4
245.2.a.d.1.2 2 56.27 even 2
245.2.e.h.116.1 4 56.19 even 6
245.2.e.h.226.1 4 56.3 even 6
245.2.e.i.116.1 4 56.51 odd 6
245.2.e.i.226.1 4 56.11 odd 6
315.2.a.e.1.1 2 24.11 even 2
560.2.a.i.1.2 2 8.5 even 2
1225.2.a.s.1.1 2 280.139 even 2
1225.2.b.f.99.2 4 280.83 odd 4
1225.2.b.f.99.3 4 280.27 odd 4
1575.2.a.p.1.2 2 120.59 even 2
1575.2.d.e.1324.2 4 120.107 odd 4
1575.2.d.e.1324.3 4 120.83 odd 4
2205.2.a.x.1.1 2 168.83 odd 2
2240.2.a.bd.1.1 2 1.1 even 1 trivial
2240.2.a.bh.1.2 2 4.3 odd 2
2800.2.a.bi.1.1 2 40.29 even 2
2800.2.g.t.449.1 4 40.37 odd 4
2800.2.g.t.449.4 4 40.13 odd 4
3920.2.a.bs.1.1 2 56.13 odd 2
4235.2.a.m.1.1 2 88.43 even 2
5040.2.a.bt.1.2 2 24.5 odd 2
5915.2.a.l.1.1 2 104.51 odd 2