# Properties

 Label 9408.2.a.cx.1.1 Level $9408$ Weight $2$ Character 9408.1 Self dual yes Analytic conductor $75.123$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9408 = 2^{6} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9408.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$75.1232582216$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 9408.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +2.00000 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +2.00000 q^{5} +1.00000 q^{9} -2.00000 q^{11} +3.00000 q^{13} +2.00000 q^{15} +8.00000 q^{17} +1.00000 q^{19} +8.00000 q^{23} -1.00000 q^{25} +1.00000 q^{27} -4.00000 q^{29} +3.00000 q^{31} -2.00000 q^{33} +1.00000 q^{37} +3.00000 q^{39} +6.00000 q^{41} -11.0000 q^{43} +2.00000 q^{45} +6.00000 q^{47} +8.00000 q^{51} +12.0000 q^{53} -4.00000 q^{55} +1.00000 q^{57} -4.00000 q^{59} +6.00000 q^{61} +6.00000 q^{65} -13.0000 q^{67} +8.00000 q^{69} -10.0000 q^{71} -11.0000 q^{73} -1.00000 q^{75} -3.00000 q^{79} +1.00000 q^{81} -2.00000 q^{83} +16.0000 q^{85} -4.00000 q^{87} +3.00000 q^{93} +2.00000 q^{95} +10.0000 q^{97} -2.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ 3.00000 0.832050 0.416025 0.909353i $$-0.363423\pi$$
0.416025 + 0.909353i $$0.363423\pi$$
$$14$$ 0 0
$$15$$ 2.00000 0.516398
$$16$$ 0 0
$$17$$ 8.00000 1.94029 0.970143 0.242536i $$-0.0779791\pi$$
0.970143 + 0.242536i $$0.0779791\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 0 0
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ 0 0
$$33$$ −2.00000 −0.348155
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 1.00000 0.164399 0.0821995 0.996616i $$-0.473806\pi$$
0.0821995 + 0.996616i $$0.473806\pi$$
$$38$$ 0 0
$$39$$ 3.00000 0.480384
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ −11.0000 −1.67748 −0.838742 0.544529i $$-0.816708\pi$$
−0.838742 + 0.544529i $$0.816708\pi$$
$$44$$ 0 0
$$45$$ 2.00000 0.298142
$$46$$ 0 0
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 8.00000 1.12022
$$52$$ 0 0
$$53$$ 12.0000 1.64833 0.824163 0.566352i $$-0.191646\pi$$
0.824163 + 0.566352i $$0.191646\pi$$
$$54$$ 0 0
$$55$$ −4.00000 −0.539360
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ 0 0
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 6.00000 0.744208
$$66$$ 0 0
$$67$$ −13.0000 −1.58820 −0.794101 0.607785i $$-0.792058\pi$$
−0.794101 + 0.607785i $$0.792058\pi$$
$$68$$ 0 0
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ −10.0000 −1.18678 −0.593391 0.804914i $$-0.702211\pi$$
−0.593391 + 0.804914i $$0.702211\pi$$
$$72$$ 0 0
$$73$$ −11.0000 −1.28745 −0.643726 0.765256i $$-0.722612\pi$$
−0.643726 + 0.765256i $$0.722612\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −3.00000 −0.337526 −0.168763 0.985657i $$-0.553977\pi$$
−0.168763 + 0.985657i $$0.553977\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −2.00000 −0.219529 −0.109764 0.993958i $$-0.535010\pi$$
−0.109764 + 0.993958i $$0.535010\pi$$
$$84$$ 0 0
$$85$$ 16.0000 1.73544
$$86$$ 0 0
$$87$$ −4.00000 −0.428845
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 3.00000 0.311086
$$94$$ 0 0
$$95$$ 2.00000 0.205196
$$96$$ 0 0
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 0 0
$$103$$ 11.0000 1.08386 0.541931 0.840423i $$-0.317693\pi$$
0.541931 + 0.840423i $$0.317693\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 11.0000 1.05361 0.526804 0.849987i $$-0.323390\pi$$
0.526804 + 0.849987i $$0.323390\pi$$
$$110$$ 0 0
$$111$$ 1.00000 0.0949158
$$112$$ 0 0
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 0 0
$$115$$ 16.0000 1.49201
$$116$$ 0 0
$$117$$ 3.00000 0.277350
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 6.00000 0.541002
$$124$$ 0 0
$$125$$ −12.0000 −1.07331
$$126$$ 0 0
$$127$$ 3.00000 0.266207 0.133103 0.991102i $$-0.457506\pi$$
0.133103 + 0.991102i $$0.457506\pi$$
$$128$$ 0 0
$$129$$ −11.0000 −0.968496
$$130$$ 0 0
$$131$$ 2.00000 0.174741 0.0873704 0.996176i $$-0.472154\pi$$
0.0873704 + 0.996176i $$0.472154\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 2.00000 0.172133
$$136$$ 0 0
$$137$$ 4.00000 0.341743 0.170872 0.985293i $$-0.445342\pi$$
0.170872 + 0.985293i $$0.445342\pi$$
$$138$$ 0 0
$$139$$ 5.00000 0.424094 0.212047 0.977259i $$-0.431987\pi$$
0.212047 + 0.977259i $$0.431987\pi$$
$$140$$ 0 0
$$141$$ 6.00000 0.505291
$$142$$ 0 0
$$143$$ −6.00000 −0.501745
$$144$$ 0 0
$$145$$ −8.00000 −0.664364
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 12.0000 0.983078 0.491539 0.870855i $$-0.336434\pi$$
0.491539 + 0.870855i $$0.336434\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ 8.00000 0.646762
$$154$$ 0 0
$$155$$ 6.00000 0.481932
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 0 0
$$159$$ 12.0000 0.951662
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 0 0
$$165$$ −4.00000 −0.311400
$$166$$ 0 0
$$167$$ −2.00000 −0.154765 −0.0773823 0.997001i $$-0.524656\pi$$
−0.0773823 + 0.997001i $$0.524656\pi$$
$$168$$ 0 0
$$169$$ −4.00000 −0.307692
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 0 0
$$173$$ −16.0000 −1.21646 −0.608229 0.793762i $$-0.708120\pi$$
−0.608229 + 0.793762i $$0.708120\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −4.00000 −0.300658
$$178$$ 0 0
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ 0 0
$$181$$ 15.0000 1.11494 0.557471 0.830197i $$-0.311772\pi$$
0.557471 + 0.830197i $$0.311772\pi$$
$$182$$ 0 0
$$183$$ 6.00000 0.443533
$$184$$ 0 0
$$185$$ 2.00000 0.147043
$$186$$ 0 0
$$187$$ −16.0000 −1.17004
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 6.00000 0.434145 0.217072 0.976156i $$-0.430349\pi$$
0.217072 + 0.976156i $$0.430349\pi$$
$$192$$ 0 0
$$193$$ 11.0000 0.791797 0.395899 0.918294i $$-0.370433\pi$$
0.395899 + 0.918294i $$0.370433\pi$$
$$194$$ 0 0
$$195$$ 6.00000 0.429669
$$196$$ 0 0
$$197$$ −8.00000 −0.569976 −0.284988 0.958531i $$-0.591990\pi$$
−0.284988 + 0.958531i $$0.591990\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 0 0
$$201$$ −13.0000 −0.916949
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 12.0000 0.838116
$$206$$ 0 0
$$207$$ 8.00000 0.556038
$$208$$ 0 0
$$209$$ −2.00000 −0.138343
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 0 0
$$213$$ −10.0000 −0.685189
$$214$$ 0 0
$$215$$ −22.0000 −1.50039
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −11.0000 −0.743311
$$220$$ 0 0
$$221$$ 24.0000 1.61441
$$222$$ 0 0
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ 0 0
$$227$$ 18.0000 1.19470 0.597351 0.801980i $$-0.296220\pi$$
0.597351 + 0.801980i $$0.296220\pi$$
$$228$$ 0 0
$$229$$ −1.00000 −0.0660819 −0.0330409 0.999454i $$-0.510519\pi$$
−0.0330409 + 0.999454i $$0.510519\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 14.0000 0.917170 0.458585 0.888650i $$-0.348356\pi$$
0.458585 + 0.888650i $$0.348356\pi$$
$$234$$ 0 0
$$235$$ 12.0000 0.782794
$$236$$ 0 0
$$237$$ −3.00000 −0.194871
$$238$$ 0 0
$$239$$ 18.0000 1.16432 0.582162 0.813073i $$-0.302207\pi$$
0.582162 + 0.813073i $$0.302207\pi$$
$$240$$ 0 0
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 3.00000 0.190885
$$248$$ 0 0
$$249$$ −2.00000 −0.126745
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ −16.0000 −1.00591
$$254$$ 0 0
$$255$$ 16.0000 1.00196
$$256$$ 0 0
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −4.00000 −0.247594
$$262$$ 0 0
$$263$$ 12.0000 0.739952 0.369976 0.929041i $$-0.379366\pi$$
0.369976 + 0.929041i $$0.379366\pi$$
$$264$$ 0 0
$$265$$ 24.0000 1.47431
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 2.00000 0.121942 0.0609711 0.998140i $$-0.480580\pi$$
0.0609711 + 0.998140i $$0.480580\pi$$
$$270$$ 0 0
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 2.00000 0.120605
$$276$$ 0 0
$$277$$ −17.0000 −1.02143 −0.510716 0.859750i $$-0.670619\pi$$
−0.510716 + 0.859750i $$0.670619\pi$$
$$278$$ 0 0
$$279$$ 3.00000 0.179605
$$280$$ 0 0
$$281$$ 20.0000 1.19310 0.596550 0.802576i $$-0.296538\pi$$
0.596550 + 0.802576i $$0.296538\pi$$
$$282$$ 0 0
$$283$$ −19.0000 −1.12943 −0.564716 0.825285i $$-0.691014\pi$$
−0.564716 + 0.825285i $$0.691014\pi$$
$$284$$ 0 0
$$285$$ 2.00000 0.118470
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 47.0000 2.76471
$$290$$ 0 0
$$291$$ 10.0000 0.586210
$$292$$ 0 0
$$293$$ 24.0000 1.40209 0.701047 0.713115i $$-0.252716\pi$$
0.701047 + 0.713115i $$0.252716\pi$$
$$294$$ 0 0
$$295$$ −8.00000 −0.465778
$$296$$ 0 0
$$297$$ −2.00000 −0.116052
$$298$$ 0 0
$$299$$ 24.0000 1.38796
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −10.0000 −0.574485
$$304$$ 0 0
$$305$$ 12.0000 0.687118
$$306$$ 0 0
$$307$$ 23.0000 1.31268 0.656340 0.754466i $$-0.272104\pi$$
0.656340 + 0.754466i $$0.272104\pi$$
$$308$$ 0 0
$$309$$ 11.0000 0.625768
$$310$$ 0 0
$$311$$ −2.00000 −0.113410 −0.0567048 0.998391i $$-0.518059\pi$$
−0.0567048 + 0.998391i $$0.518059\pi$$
$$312$$ 0 0
$$313$$ −17.0000 −0.960897 −0.480448 0.877023i $$-0.659526\pi$$
−0.480448 + 0.877023i $$0.659526\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 24.0000 1.34797 0.673987 0.738743i $$-0.264580\pi$$
0.673987 + 0.738743i $$0.264580\pi$$
$$318$$ 0 0
$$319$$ 8.00000 0.447914
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 8.00000 0.445132
$$324$$ 0 0
$$325$$ −3.00000 −0.166410
$$326$$ 0 0
$$327$$ 11.0000 0.608301
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −17.0000 −0.934405 −0.467202 0.884150i $$-0.654738\pi$$
−0.467202 + 0.884150i $$0.654738\pi$$
$$332$$ 0 0
$$333$$ 1.00000 0.0547997
$$334$$ 0 0
$$335$$ −26.0000 −1.42053
$$336$$ 0 0
$$337$$ 21.0000 1.14394 0.571971 0.820274i $$-0.306179\pi$$
0.571971 + 0.820274i $$0.306179\pi$$
$$338$$ 0 0
$$339$$ −14.0000 −0.760376
$$340$$ 0 0
$$341$$ −6.00000 −0.324918
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 16.0000 0.861411
$$346$$ 0 0
$$347$$ −24.0000 −1.28839 −0.644194 0.764862i $$-0.722807\pi$$
−0.644194 + 0.764862i $$0.722807\pi$$
$$348$$ 0 0
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ 3.00000 0.160128
$$352$$ 0 0
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ −20.0000 −1.06149
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −20.0000 −1.05556 −0.527780 0.849381i $$-0.676975\pi$$
−0.527780 + 0.849381i $$0.676975\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 0 0
$$363$$ −7.00000 −0.367405
$$364$$ 0 0
$$365$$ −22.0000 −1.15153
$$366$$ 0 0
$$367$$ 5.00000 0.260998 0.130499 0.991448i $$-0.458342\pi$$
0.130499 + 0.991448i $$0.458342\pi$$
$$368$$ 0 0
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 5.00000 0.258890 0.129445 0.991587i $$-0.458680\pi$$
0.129445 + 0.991587i $$0.458680\pi$$
$$374$$ 0 0
$$375$$ −12.0000 −0.619677
$$376$$ 0 0
$$377$$ −12.0000 −0.618031
$$378$$ 0 0
$$379$$ −13.0000 −0.667765 −0.333883 0.942615i $$-0.608359\pi$$
−0.333883 + 0.942615i $$0.608359\pi$$
$$380$$ 0 0
$$381$$ 3.00000 0.153695
$$382$$ 0 0
$$383$$ −28.0000 −1.43073 −0.715367 0.698749i $$-0.753740\pi$$
−0.715367 + 0.698749i $$0.753740\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −11.0000 −0.559161
$$388$$ 0 0
$$389$$ −10.0000 −0.507020 −0.253510 0.967333i $$-0.581585\pi$$
−0.253510 + 0.967333i $$0.581585\pi$$
$$390$$ 0 0
$$391$$ 64.0000 3.23662
$$392$$ 0 0
$$393$$ 2.00000 0.100887
$$394$$ 0 0
$$395$$ −6.00000 −0.301893
$$396$$ 0 0
$$397$$ −3.00000 −0.150566 −0.0752828 0.997162i $$-0.523986\pi$$
−0.0752828 + 0.997162i $$0.523986\pi$$
$$398$$ 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
$$403$$ 9.00000 0.448322
$$404$$ 0 0
$$405$$ 2.00000 0.0993808
$$406$$ 0 0
$$407$$ −2.00000 −0.0991363
$$408$$ 0 0
$$409$$ −19.0000 −0.939490 −0.469745 0.882802i $$-0.655654\pi$$
−0.469745 + 0.882802i $$0.655654\pi$$
$$410$$ 0 0
$$411$$ 4.00000 0.197305
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −4.00000 −0.196352
$$416$$ 0 0
$$417$$ 5.00000 0.244851
$$418$$ 0 0
$$419$$ −18.0000 −0.879358 −0.439679 0.898155i $$-0.644908\pi$$
−0.439679 + 0.898155i $$0.644908\pi$$
$$420$$ 0 0
$$421$$ 27.0000 1.31590 0.657950 0.753062i $$-0.271424\pi$$
0.657950 + 0.753062i $$0.271424\pi$$
$$422$$ 0 0
$$423$$ 6.00000 0.291730
$$424$$ 0 0
$$425$$ −8.00000 −0.388057
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −6.00000 −0.289683
$$430$$ 0 0
$$431$$ −30.0000 −1.44505 −0.722525 0.691345i $$-0.757018\pi$$
−0.722525 + 0.691345i $$0.757018\pi$$
$$432$$ 0 0
$$433$$ −25.0000 −1.20142 −0.600712 0.799466i $$-0.705116\pi$$
−0.600712 + 0.799466i $$0.705116\pi$$
$$434$$ 0 0
$$435$$ −8.00000 −0.383571
$$436$$ 0 0
$$437$$ 8.00000 0.382692
$$438$$ 0 0
$$439$$ 24.0000 1.14546 0.572729 0.819745i $$-0.305885\pi$$
0.572729 + 0.819745i $$0.305885\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 4.00000 0.190046 0.0950229 0.995475i $$-0.469708\pi$$
0.0950229 + 0.995475i $$0.469708\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 12.0000 0.567581
$$448$$ 0 0
$$449$$ 22.0000 1.03824 0.519122 0.854700i $$-0.326259\pi$$
0.519122 + 0.854700i $$0.326259\pi$$
$$450$$ 0 0
$$451$$ −12.0000 −0.565058
$$452$$ 0 0
$$453$$ −8.00000 −0.375873
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 13.0000 0.608114 0.304057 0.952654i $$-0.401659\pi$$
0.304057 + 0.952654i $$0.401659\pi$$
$$458$$ 0 0
$$459$$ 8.00000 0.373408
$$460$$ 0 0
$$461$$ −4.00000 −0.186299 −0.0931493 0.995652i $$-0.529693\pi$$
−0.0931493 + 0.995652i $$0.529693\pi$$
$$462$$ 0 0
$$463$$ −11.0000 −0.511213 −0.255607 0.966781i $$-0.582275\pi$$
−0.255607 + 0.966781i $$0.582275\pi$$
$$464$$ 0 0
$$465$$ 6.00000 0.278243
$$466$$ 0 0
$$467$$ −34.0000 −1.57333 −0.786666 0.617379i $$-0.788195\pi$$
−0.786666 + 0.617379i $$0.788195\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −2.00000 −0.0921551
$$472$$ 0 0
$$473$$ 22.0000 1.01156
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 0 0
$$477$$ 12.0000 0.549442
$$478$$ 0 0
$$479$$ −28.0000 −1.27935 −0.639676 0.768644i $$-0.720932\pi$$
−0.639676 + 0.768644i $$0.720932\pi$$
$$480$$ 0 0
$$481$$ 3.00000 0.136788
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 20.0000 0.908153
$$486$$ 0 0
$$487$$ −19.0000 −0.860972 −0.430486 0.902597i $$-0.641658\pi$$
−0.430486 + 0.902597i $$0.641658\pi$$
$$488$$ 0 0
$$489$$ 4.00000 0.180886
$$490$$ 0 0
$$491$$ 36.0000 1.62466 0.812329 0.583200i $$-0.198200\pi$$
0.812329 + 0.583200i $$0.198200\pi$$
$$492$$ 0 0
$$493$$ −32.0000 −1.44121
$$494$$ 0 0
$$495$$ −4.00000 −0.179787
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 29.0000 1.29822 0.649109 0.760695i $$-0.275142\pi$$
0.649109 + 0.760695i $$0.275142\pi$$
$$500$$ 0 0
$$501$$ −2.00000 −0.0893534
$$502$$ 0 0
$$503$$ −30.0000 −1.33763 −0.668817 0.743427i $$-0.733199\pi$$
−0.668817 + 0.743427i $$0.733199\pi$$
$$504$$ 0 0
$$505$$ −20.0000 −0.889988
$$506$$ 0 0
$$507$$ −4.00000 −0.177646
$$508$$ 0 0
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 1.00000 0.0441511
$$514$$ 0 0
$$515$$ 22.0000 0.969436
$$516$$ 0 0
$$517$$ −12.0000 −0.527759
$$518$$ 0 0
$$519$$ −16.0000 −0.702322
$$520$$ 0 0
$$521$$ −36.0000 −1.57719 −0.788594 0.614914i $$-0.789191\pi$$
−0.788594 + 0.614914i $$0.789191\pi$$
$$522$$ 0 0
$$523$$ 31.0000 1.35554 0.677768 0.735276i $$-0.262948\pi$$
0.677768 + 0.735276i $$0.262948\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 24.0000 1.04546
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ −4.00000 −0.173585
$$532$$ 0 0
$$533$$ 18.0000 0.779667
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −6.00000 −0.258919
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 15.0000 0.644900 0.322450 0.946586i $$-0.395494\pi$$
0.322450 + 0.946586i $$0.395494\pi$$
$$542$$ 0 0
$$543$$ 15.0000 0.643712
$$544$$ 0 0
$$545$$ 22.0000 0.942376
$$546$$ 0 0
$$547$$ 12.0000 0.513083 0.256541 0.966533i $$-0.417417\pi$$
0.256541 + 0.966533i $$0.417417\pi$$
$$548$$ 0 0
$$549$$ 6.00000 0.256074
$$550$$ 0 0
$$551$$ −4.00000 −0.170406
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 2.00000 0.0848953
$$556$$ 0 0
$$557$$ −22.0000 −0.932170 −0.466085 0.884740i $$-0.654336\pi$$
−0.466085 + 0.884740i $$0.654336\pi$$
$$558$$ 0 0
$$559$$ −33.0000 −1.39575
$$560$$ 0 0
$$561$$ −16.0000 −0.675521
$$562$$ 0 0
$$563$$ 46.0000 1.93867 0.969334 0.245745i $$-0.0790327\pi$$
0.969334 + 0.245745i $$0.0790327\pi$$
$$564$$ 0 0
$$565$$ −28.0000 −1.17797
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 21.0000 0.878823 0.439411 0.898286i $$-0.355187\pi$$
0.439411 + 0.898286i $$0.355187\pi$$
$$572$$ 0 0
$$573$$ 6.00000 0.250654
$$574$$ 0 0
$$575$$ −8.00000 −0.333623
$$576$$ 0 0
$$577$$ −41.0000 −1.70685 −0.853426 0.521214i $$-0.825479\pi$$
−0.853426 + 0.521214i $$0.825479\pi$$
$$578$$ 0 0
$$579$$ 11.0000 0.457144
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −24.0000 −0.993978
$$584$$ 0 0
$$585$$ 6.00000 0.248069
$$586$$ 0 0
$$587$$ −32.0000 −1.32078 −0.660391 0.750922i $$-0.729609\pi$$
−0.660391 + 0.750922i $$0.729609\pi$$
$$588$$ 0 0
$$589$$ 3.00000 0.123613
$$590$$ 0 0
$$591$$ −8.00000 −0.329076
$$592$$ 0 0
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 8.00000 0.327418
$$598$$ 0 0
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ −1.00000 −0.0407909 −0.0203954 0.999792i $$-0.506493\pi$$
−0.0203954 + 0.999792i $$0.506493\pi$$
$$602$$ 0 0
$$603$$ −13.0000 −0.529401
$$604$$ 0 0
$$605$$ −14.0000 −0.569181
$$606$$ 0 0
$$607$$ −3.00000 −0.121766 −0.0608831 0.998145i $$-0.519392\pi$$
−0.0608831 + 0.998145i $$0.519392\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 18.0000 0.728202
$$612$$ 0 0
$$613$$ 30.0000 1.21169 0.605844 0.795583i $$-0.292835\pi$$
0.605844 + 0.795583i $$0.292835\pi$$
$$614$$ 0 0
$$615$$ 12.0000 0.483887
$$616$$ 0 0
$$617$$ 26.0000 1.04672 0.523360 0.852111i $$-0.324678\pi$$
0.523360 + 0.852111i $$0.324678\pi$$
$$618$$ 0 0
$$619$$ 11.0000 0.442127 0.221064 0.975259i $$-0.429047\pi$$
0.221064 + 0.975259i $$0.429047\pi$$
$$620$$ 0 0
$$621$$ 8.00000 0.321029
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0 0
$$627$$ −2.00000 −0.0798723
$$628$$ 0 0
$$629$$ 8.00000 0.318981
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 0 0
$$633$$ 4.00000 0.158986
$$634$$ 0 0
$$635$$ 6.00000 0.238103
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −10.0000 −0.395594
$$640$$ 0 0
$$641$$ −40.0000 −1.57991 −0.789953 0.613168i $$-0.789895\pi$$
−0.789953 + 0.613168i $$0.789895\pi$$
$$642$$ 0 0
$$643$$ −35.0000 −1.38027 −0.690133 0.723683i $$-0.742448\pi$$
−0.690133 + 0.723683i $$0.742448\pi$$
$$644$$ 0 0
$$645$$ −22.0000 −0.866249
$$646$$ 0 0
$$647$$ 6.00000 0.235884 0.117942 0.993020i $$-0.462370\pi$$
0.117942 + 0.993020i $$0.462370\pi$$
$$648$$ 0 0
$$649$$ 8.00000 0.314027
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 6.00000 0.234798 0.117399 0.993085i $$-0.462544\pi$$
0.117399 + 0.993085i $$0.462544\pi$$
$$654$$ 0 0
$$655$$ 4.00000 0.156293
$$656$$ 0 0
$$657$$ −11.0000 −0.429151
$$658$$ 0 0
$$659$$ 28.0000 1.09073 0.545363 0.838200i $$-0.316392\pi$$
0.545363 + 0.838200i $$0.316392\pi$$
$$660$$ 0 0
$$661$$ 29.0000 1.12797 0.563985 0.825785i $$-0.309268\pi$$
0.563985 + 0.825785i $$0.309268\pi$$
$$662$$ 0 0
$$663$$ 24.0000 0.932083
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −32.0000 −1.23904
$$668$$ 0 0
$$669$$ 8.00000 0.309298
$$670$$ 0 0
$$671$$ −12.0000 −0.463255
$$672$$ 0 0
$$673$$ −1.00000 −0.0385472 −0.0192736 0.999814i $$-0.506135\pi$$
−0.0192736 + 0.999814i $$0.506135\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ −12.0000 −0.461197 −0.230599 0.973049i $$-0.574068\pi$$
−0.230599 + 0.973049i $$0.574068\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 18.0000 0.689761
$$682$$ 0 0
$$683$$ −36.0000 −1.37750 −0.688751 0.724998i $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$684$$ 0 0
$$685$$ 8.00000 0.305664
$$686$$ 0 0
$$687$$ −1.00000 −0.0381524
$$688$$ 0 0
$$689$$ 36.0000 1.37149
$$690$$ 0 0
$$691$$ 43.0000 1.63580 0.817899 0.575362i $$-0.195139\pi$$
0.817899 + 0.575362i $$0.195139\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 10.0000 0.379322
$$696$$ 0 0
$$697$$ 48.0000 1.81813
$$698$$ 0 0
$$699$$ 14.0000 0.529529
$$700$$ 0 0
$$701$$ 8.00000 0.302156 0.151078 0.988522i $$-0.451726\pi$$
0.151078 + 0.988522i $$0.451726\pi$$
$$702$$ 0 0
$$703$$ 1.00000 0.0377157
$$704$$ 0 0
$$705$$ 12.0000 0.451946
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −14.0000 −0.525781 −0.262891 0.964826i $$-0.584676\pi$$
−0.262891 + 0.964826i $$0.584676\pi$$
$$710$$ 0 0
$$711$$ −3.00000 −0.112509
$$712$$ 0 0
$$713$$ 24.0000 0.898807
$$714$$ 0 0
$$715$$ −12.0000 −0.448775
$$716$$ 0 0
$$717$$ 18.0000 0.672222
$$718$$ 0 0
$$719$$ −6.00000 −0.223762 −0.111881 0.993722i $$-0.535688\pi$$
−0.111881 + 0.993722i $$0.535688\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 14.0000 0.520666
$$724$$ 0 0
$$725$$ 4.00000 0.148556
$$726$$ 0 0
$$727$$ −23.0000 −0.853023 −0.426511 0.904482i $$-0.640258\pi$$
−0.426511 + 0.904482i $$0.640258\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −88.0000 −3.25480
$$732$$ 0 0
$$733$$ −45.0000 −1.66211 −0.831056 0.556188i $$-0.812263\pi$$
−0.831056 + 0.556188i $$0.812263\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 26.0000 0.957722
$$738$$ 0 0
$$739$$ 9.00000 0.331070 0.165535 0.986204i $$-0.447065\pi$$
0.165535 + 0.986204i $$0.447065\pi$$
$$740$$ 0 0
$$741$$ 3.00000 0.110208
$$742$$ 0 0
$$743$$ −18.0000 −0.660356 −0.330178 0.943919i $$-0.607109\pi$$
−0.330178 + 0.943919i $$0.607109\pi$$
$$744$$ 0 0
$$745$$ 24.0000 0.879292
$$746$$ 0 0
$$747$$ −2.00000 −0.0731762
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 15.0000 0.547358 0.273679 0.961821i $$-0.411759\pi$$
0.273679 + 0.961821i $$0.411759\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −16.0000 −0.582300
$$756$$ 0 0
$$757$$ −42.0000 −1.52652 −0.763258 0.646094i $$-0.776401\pi$$
−0.763258 + 0.646094i $$0.776401\pi$$
$$758$$ 0 0
$$759$$ −16.0000 −0.580763
$$760$$ 0 0
$$761$$ 8.00000 0.290000 0.145000 0.989432i $$-0.453682\pi$$
0.145000 + 0.989432i $$0.453682\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 16.0000 0.578481
$$766$$ 0 0
$$767$$ −12.0000 −0.433295
$$768$$ 0 0
$$769$$ 31.0000 1.11789 0.558944 0.829205i $$-0.311207\pi$$
0.558944 + 0.829205i $$0.311207\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ 0 0
$$773$$ −22.0000 −0.791285 −0.395643 0.918405i $$-0.629478\pi$$
−0.395643 + 0.918405i $$0.629478\pi$$
$$774$$ 0 0
$$775$$ −3.00000 −0.107763
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 6.00000 0.214972
$$780$$ 0 0
$$781$$ 20.0000 0.715656
$$782$$ 0 0
$$783$$ −4.00000 −0.142948
$$784$$ 0 0
$$785$$ −4.00000 −0.142766
$$786$$ 0 0
$$787$$ −24.0000 −0.855508 −0.427754 0.903895i $$-0.640695\pi$$
−0.427754 + 0.903895i $$0.640695\pi$$
$$788$$ 0 0
$$789$$ 12.0000 0.427211
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 18.0000 0.639199
$$794$$ 0 0
$$795$$ 24.0000 0.851192
$$796$$ 0 0
$$797$$ 48.0000 1.70025 0.850124 0.526583i $$-0.176527\pi$$
0.850124 + 0.526583i $$0.176527\pi$$
$$798$$ 0 0
$$799$$ 48.0000 1.69812
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 22.0000 0.776363
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 2.00000 0.0704033
$$808$$ 0 0
$$809$$ 22.0000 0.773479 0.386739 0.922189i $$-0.373601\pi$$
0.386739 + 0.922189i $$0.373601\pi$$
$$810$$ 0 0
$$811$$ 32.0000 1.12367 0.561836 0.827249i $$-0.310095\pi$$
0.561836 + 0.827249i $$0.310095\pi$$
$$812$$ 0 0
$$813$$ −24.0000 −0.841717
$$814$$ 0 0
$$815$$ 8.00000 0.280228
$$816$$ 0 0
$$817$$ −11.0000 −0.384841
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −2.00000 −0.0698005 −0.0349002 0.999391i $$-0.511111\pi$$
−0.0349002 + 0.999391i $$0.511111\pi$$
$$822$$ 0 0
$$823$$ 40.0000 1.39431 0.697156 0.716919i $$-0.254448\pi$$
0.697156 + 0.716919i $$0.254448\pi$$
$$824$$ 0 0
$$825$$ 2.00000 0.0696311
$$826$$ 0 0
$$827$$ −54.0000 −1.87776 −0.938882 0.344239i $$-0.888137\pi$$
−0.938882 + 0.344239i $$0.888137\pi$$
$$828$$ 0 0
$$829$$ 11.0000 0.382046 0.191023 0.981586i $$-0.438820\pi$$
0.191023 + 0.981586i $$0.438820\pi$$
$$830$$ 0 0
$$831$$ −17.0000 −0.589723
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −4.00000 −0.138426
$$836$$ 0 0
$$837$$ 3.00000 0.103695
$$838$$ 0 0
$$839$$ −4.00000 −0.138095 −0.0690477 0.997613i $$-0.521996\pi$$
−0.0690477 + 0.997613i $$0.521996\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 0 0
$$843$$ 20.0000 0.688837
$$844$$ 0 0
$$845$$ −8.00000 −0.275208
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −19.0000 −0.652078
$$850$$ 0 0
$$851$$ 8.00000 0.274236
$$852$$ 0 0
$$853$$ −23.0000 −0.787505 −0.393753 0.919216i $$-0.628823\pi$$
−0.393753 + 0.919216i $$0.628823\pi$$
$$854$$ 0 0
$$855$$ 2.00000 0.0683986
$$856$$ 0 0
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 0 0
$$859$$ −8.00000 −0.272956 −0.136478 0.990643i $$-0.543578\pi$$
−0.136478 + 0.990643i $$0.543578\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 46.0000 1.56586 0.782929 0.622111i $$-0.213725\pi$$
0.782929 + 0.622111i $$0.213725\pi$$
$$864$$ 0 0
$$865$$ −32.0000 −1.08803
$$866$$ 0 0
$$867$$ 47.0000 1.59620
$$868$$ 0 0
$$869$$ 6.00000 0.203536
$$870$$ 0 0
$$871$$ −39.0000 −1.32146
$$872$$ 0 0
$$873$$ 10.0000 0.338449
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 38.0000 1.28317 0.641584 0.767052i $$-0.278277\pi$$
0.641584 + 0.767052i $$0.278277\pi$$
$$878$$ 0 0
$$879$$ 24.0000 0.809500
$$880$$ 0 0
$$881$$ 48.0000 1.61716 0.808581 0.588386i $$-0.200236\pi$$
0.808581 + 0.588386i $$0.200236\pi$$
$$882$$ 0 0
$$883$$ −29.0000 −0.975928 −0.487964 0.872864i $$-0.662260\pi$$
−0.487964 + 0.872864i $$0.662260\pi$$
$$884$$ 0 0
$$885$$ −8.00000 −0.268917
$$886$$ 0 0
$$887$$ −6.00000 −0.201460 −0.100730 0.994914i $$-0.532118\pi$$
−0.100730 + 0.994914i $$0.532118\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −2.00000 −0.0670025
$$892$$ 0 0
$$893$$ 6.00000 0.200782
$$894$$ 0 0
$$895$$ −12.0000 −0.401116
$$896$$ 0 0
$$897$$ 24.0000 0.801337
$$898$$ 0 0
$$899$$ −12.0000 −0.400222
$$900$$ 0 0
$$901$$ 96.0000 3.19822
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 30.0000 0.997234
$$906$$ 0 0
$$907$$ −21.0000 −0.697294 −0.348647 0.937254i $$-0.613359\pi$$
−0.348647 + 0.937254i $$0.613359\pi$$
$$908$$ 0 0
$$909$$ −10.0000 −0.331679
$$910$$ 0 0
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ 0 0
$$913$$ 4.00000 0.132381
$$914$$ 0 0
$$915$$ 12.0000 0.396708
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −11.0000 −0.362857 −0.181428 0.983404i $$-0.558072\pi$$
−0.181428 + 0.983404i $$0.558072\pi$$
$$920$$ 0 0
$$921$$ 23.0000 0.757876
$$922$$ 0 0
$$923$$ −30.0000 −0.987462
$$924$$ 0 0
$$925$$ −1.00000 −0.0328798
$$926$$ 0 0
$$927$$ 11.0000 0.361287
$$928$$ 0 0
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −2.00000 −0.0654771
$$934$$ 0 0
$$935$$ −32.0000 −1.04651
$$936$$ 0 0
$$937$$ −49.0000 −1.60076 −0.800380 0.599493i $$-0.795369\pi$$
−0.800380 + 0.599493i $$0.795369\pi$$
$$938$$ 0 0
$$939$$ −17.0000 −0.554774
$$940$$ 0 0
$$941$$ −52.0000 −1.69515 −0.847576 0.530674i $$-0.821939\pi$$
−0.847576 + 0.530674i $$0.821939\pi$$
$$942$$ 0 0
$$943$$ 48.0000 1.56310
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −10.0000 −0.324956 −0.162478 0.986712i $$-0.551949\pi$$
−0.162478 + 0.986712i $$0.551949\pi$$
$$948$$ 0 0
$$949$$ −33.0000 −1.07123
$$950$$ 0 0
$$951$$ 24.0000 0.778253
$$952$$ 0 0
$$953$$ −28.0000 −0.907009 −0.453504 0.891254i $$-0.649826\pi$$
−0.453504 + 0.891254i $$0.649826\pi$$
$$954$$ 0 0
$$955$$ 12.0000 0.388311
$$956$$ 0 0
$$957$$ 8.00000 0.258603
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 22.0000 0.708205
$$966$$ 0 0
$$967$$ −31.0000 −0.996893 −0.498446 0.866921i $$-0.666096\pi$$
−0.498446 + 0.866921i $$0.666096\pi$$
$$968$$ 0 0
$$969$$ 8.00000 0.256997
$$970$$ 0 0
$$971$$ −36.0000 −1.15529 −0.577647 0.816286i $$-0.696029\pi$$
−0.577647 + 0.816286i $$0.696029\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −3.00000 −0.0960769
$$976$$ 0 0
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 11.0000 0.351203
$$982$$ 0 0
$$983$$ −12.0000 −0.382741 −0.191370 0.981518i $$-0.561293\pi$$
−0.191370 + 0.981518i $$0.561293\pi$$
$$984$$ 0 0
$$985$$ −16.0000 −0.509802
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −88.0000 −2.79824
$$990$$ 0 0
$$991$$ 11.0000 0.349427 0.174713 0.984619i $$-0.444100\pi$$
0.174713 + 0.984619i $$0.444100\pi$$
$$992$$ 0 0
$$993$$ −17.0000 −0.539479
$$994$$ 0 0
$$995$$ 16.0000 0.507234
$$996$$ 0 0
$$997$$ 1.00000 0.0316703 0.0158352 0.999875i $$-0.494959\pi$$
0.0158352 + 0.999875i $$0.494959\pi$$
$$998$$ 0 0
$$999$$ 1.00000 0.0316386
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 9408.2.a.cx.1.1 1
4.3 odd 2 9408.2.a.bi.1.1 1
7.2 even 3 1344.2.q.b.193.1 2
7.4 even 3 1344.2.q.b.961.1 2
7.6 odd 2 9408.2.a.i.1.1 1
8.3 odd 2 2352.2.a.o.1.1 1
8.5 even 2 588.2.a.a.1.1 1
24.5 odd 2 1764.2.a.h.1.1 1
24.11 even 2 7056.2.a.bs.1.1 1
28.11 odd 6 1344.2.q.n.961.1 2
28.23 odd 6 1344.2.q.n.193.1 2
28.27 even 2 9408.2.a.bx.1.1 1
56.3 even 6 2352.2.q.q.961.1 2
56.5 odd 6 588.2.i.b.361.1 2
56.11 odd 6 336.2.q.c.289.1 2
56.13 odd 2 588.2.a.f.1.1 1
56.19 even 6 2352.2.q.q.1537.1 2
56.27 even 2 2352.2.a.k.1.1 1
56.37 even 6 84.2.i.a.25.1 2
56.45 odd 6 588.2.i.b.373.1 2
56.51 odd 6 336.2.q.c.193.1 2
56.53 even 6 84.2.i.a.37.1 yes 2
168.5 even 6 1764.2.k.j.361.1 2
168.11 even 6 1008.2.s.c.289.1 2
168.53 odd 6 252.2.k.a.37.1 2
168.83 odd 2 7056.2.a.o.1.1 1
168.101 even 6 1764.2.k.j.1549.1 2
168.107 even 6 1008.2.s.c.865.1 2
168.125 even 2 1764.2.a.c.1.1 1
168.149 odd 6 252.2.k.a.109.1 2
280.37 odd 12 2100.2.bc.a.949.1 4
280.53 odd 12 2100.2.bc.a.1549.1 4
280.93 odd 12 2100.2.bc.a.949.2 4
280.109 even 6 2100.2.q.b.1801.1 2
280.149 even 6 2100.2.q.b.1201.1 2
280.277 odd 12 2100.2.bc.a.1549.2 4
504.149 odd 6 2268.2.i.b.865.1 2
504.205 even 6 2268.2.l.b.109.1 2
504.221 odd 6 2268.2.l.g.541.1 2
504.277 even 6 2268.2.i.g.2053.1 2
504.317 odd 6 2268.2.l.g.109.1 2
504.373 even 6 2268.2.i.g.865.1 2
504.389 odd 6 2268.2.i.b.2053.1 2
504.445 even 6 2268.2.l.b.541.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
84.2.i.a.25.1 2 56.37 even 6
84.2.i.a.37.1 yes 2 56.53 even 6
252.2.k.a.37.1 2 168.53 odd 6
252.2.k.a.109.1 2 168.149 odd 6
336.2.q.c.193.1 2 56.51 odd 6
336.2.q.c.289.1 2 56.11 odd 6
588.2.a.a.1.1 1 8.5 even 2
588.2.a.f.1.1 1 56.13 odd 2
588.2.i.b.361.1 2 56.5 odd 6
588.2.i.b.373.1 2 56.45 odd 6
1008.2.s.c.289.1 2 168.11 even 6
1008.2.s.c.865.1 2 168.107 even 6
1344.2.q.b.193.1 2 7.2 even 3
1344.2.q.b.961.1 2 7.4 even 3
1344.2.q.n.193.1 2 28.23 odd 6
1344.2.q.n.961.1 2 28.11 odd 6
1764.2.a.c.1.1 1 168.125 even 2
1764.2.a.h.1.1 1 24.5 odd 2
1764.2.k.j.361.1 2 168.5 even 6
1764.2.k.j.1549.1 2 168.101 even 6
2100.2.q.b.1201.1 2 280.149 even 6
2100.2.q.b.1801.1 2 280.109 even 6
2100.2.bc.a.949.1 4 280.37 odd 12
2100.2.bc.a.949.2 4 280.93 odd 12
2100.2.bc.a.1549.1 4 280.53 odd 12
2100.2.bc.a.1549.2 4 280.277 odd 12
2268.2.i.b.865.1 2 504.149 odd 6
2268.2.i.b.2053.1 2 504.389 odd 6
2268.2.i.g.865.1 2 504.373 even 6
2268.2.i.g.2053.1 2 504.277 even 6
2268.2.l.b.109.1 2 504.205 even 6
2268.2.l.b.541.1 2 504.445 even 6
2268.2.l.g.109.1 2 504.317 odd 6
2268.2.l.g.541.1 2 504.221 odd 6
2352.2.a.k.1.1 1 56.27 even 2
2352.2.a.o.1.1 1 8.3 odd 2
2352.2.q.q.961.1 2 56.3 even 6
2352.2.q.q.1537.1 2 56.19 even 6
7056.2.a.o.1.1 1 168.83 odd 2
7056.2.a.bs.1.1 1 24.11 even 2
9408.2.a.i.1.1 1 7.6 odd 2
9408.2.a.bi.1.1 1 4.3 odd 2
9408.2.a.bx.1.1 1 28.27 even 2
9408.2.a.cx.1.1 1 1.1 even 1 trivial